02 Trasformata Di Fourier

8/19/2019 02 Trasformata Di Fourier

1/22

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S ( f ) = A! sinc f ! ( )

A

t 0.5 !

f

!"

-1 "

1"

s(t )

S ( f )


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as1(t )+ bs2(t ) ! aS 1( f )+ bS 2( f )

s(t ! t 1) " S ( f )e! j 2# ft 1

s(t )e j 2! f 1t

" S ( f # f 1)

s(t )dt " # $ j 1

2% f S ( f )

ds(t )

dt ! j 2" f S ( f )


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v(t ) = s(t ) cos ! 0t ( )

V ( f ) =1

2S ( f ! f

0)+ S ( f + f

0)

( ) -f 0

V(f)

f 0 f

f

S(f)


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V ( f ) = S ( f ) H ( f )H(f)s(t) v(t)

v(t ) = s(! ) h(

"#

+#

$ t "! ) d !


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v(t )= F "1

S ( f ) H ( f )( )= S ( f ) H ( f ) e j 2# ft

df ="$

+$

% S ( f ) h(& ) e" j 2# f &

"$

+$

% d & '

( )

*

+ , e

j 2# ft df

"$

+$

%

= h(" ) S ( f )#$

+$ % e j 2& f (t #" ) df

'

( )

*

+ ,

#$

+$ % d " = h(" ) s(t #" )

#$

+$ % d " = s(" ) h(t #" )

#$

+$ % d "

v(t ) = " (# ) h(t $ # )$%

+%

& d # = h(t )


8/22


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E = s2(t )dt

!"

+"

#

E = s2(t ) dt

!"

+"

# = S ( f )S *( f )df = S ( f )

2df

!"

+"

# !"

+"

#

G( f ) = S ( f )2 E ( f 1, f 2 ) = S ( f )

2df

f 1

f 2

! + S ( f )2df

" f 2

" f 1

!


9/22


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G0 ( f ) = V ( f )2

= H ( f )2S ( f )

2= H ( f )

2G( f )

H ( f ) = e! j 2" fT D

# E 0 = E

H ( f ) = h0 e! j 2" fT D

# v(t ) = h0 s(t !T D)


10/22


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t

v(t)

B-B

1

f

H(f)

filtro ideale

filtro reale

B-B

1

f

H(f)

1/ ! 2

s(t)

t


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- B B f

1

h(t )= H ( f ) e j 2! ft df

!"

"

# = 1 $e j 2! ft df ! B

B

# =e j 2! Bt ! e. j 2! Bt

j 2! t = ...

... =1

! t

e j 2! Bt

! e! j 2! Bt

j 2= 2 B

sin2! Bt

2! Bt = 2 B sinc 2 Bt ( )

t -1/2 B 1/2B t

"(t ) h(t )

L’uscita arriva prima dell’ingresso!


13/22


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R12(! ) = limT "#

1

T s1(t )s2 (t + ! )dt $T / 2

T / 2

%

R12(! ) =1

T 0

s1(t )s2 (t +! )dt

"T 0 / 2

T 0 / 2

#

R12(! ) = s1(t )s2 (t + ! )dt

"#

+#

$


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t

t

s1(t )=s(t )

s2(t )=s(t-! )

R12(! )= s1(t )s2(t +! )dt

!"

+"

# = s(t )s(t !" +! )dt

!"

+"

# R

12(" ) = s(t )s(t !" +" )dt

!"

+"

# = s2(t ) dt !"

+"

# = E s


15/22


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P12 = limT !"

1

T [s1(t ) + s2 (t )]

2

#T / 2

T / 2

$ dt

P12 = limT !"

1

T [s1(t ) + s2 (t + # )]

2

$T / 2

T / 2

% dt

P12 = P1+P2 +2 R12(" )


16/22


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Rs(! ) = s(t )s(t +! )

!"

"

# dt

Rs(0) = s(t )s(t +0)!"

"

# dt = s2

(t )!"

"

# dt = E s

Rs(!! ) = s(t )s(t !! )

!"

"

# dt = s( x+! )s( x)!"

"

# dt = Rs(! )

0 ! s(t )± s(t +! )( )2

"#

#

$ dt = s2(t )"#

#

$ dt + s2 (t +! )"#

#

$ dt ±2 s(t )s(t +! )"#

#

$ dt =

0 ! 2 E s ± R

s ! ( )%& '( ) E s * ! Rs ! ( ) ) E s * Rs ! ( )

Rs (0) = E s

Rs(! ) = R

s(!! )

Rs(0) " R

s(! )


17/22


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s(t ) = A rect t

W

!

"#

$

%& ' Rs(! ) = s(t )s(t (! )

()

)

* dt = A2 rect t

W

!

"#

$

%&rect

t (!

W

!

"#

$

%&

()

)

* dt

W-W

A

t

s(t)

+ o – è lo stesso!

W invece di " per non

confondersi!

Rs(! ) = A

21

!W +!

W

" dt = A2 W ! !W +! ( )#$ %&= A2 2W ! ! ( )

-W W

-W +" W+"

s(t )

s(t !! )

s t ( ) s(t !! )

-W +" W

A

A

A2


18/22


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s(t ) = A rect t

W

!

"#

$

%& ' Rs(! ) = s(t )s(t (! )

()

)

* dt = A2 rect t

W

!

"#

$

%&rect

t (!

W

!

"#

$

%&

()

)

* dt

W-W

A

t

s(t)

2AW

2W-2W !

R( !)

Rs(! )= 0

Rs(! ) =

A2 2W ! ! ( ) ! " 00 ! > 0

#$%

&%2

-W W

-W +" W+"

s(t )

s(t !! )

A

A


19/22


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! 5'/"

R(! ) =1

T 0

cm e

j 2! mt

T 0

m=!"

+"

#$

%

&&

'

(

)) cn e

j 2! n(t +! )

T 0

n=!"

+"

#$

%

&&

'

(

)) dt

!T 0 /2

T 0 /2

*

R(! ) = I mn I mn =n="#

+#

$m="#

+#

$ 1

T 0e j

2% n!

T 0 cmcn e j

2% (m+n)t

T 0 dt

"T 0 / 2

T 0 / 2

&

I mn = c

mcn e

j 2! n!

T 0sin ! (m+n)( )

! (m+n)=

1 se m+n = 0

0 se m+n ! 0

"#$

%$


22/22


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02 Trasformata Di Fourier